How can I smooth a triangle fan?
The Subsurf Modifier doesn't work well with Triangles.

Ideal topology consists of mainly quads (faces with four vertices) and each vertex is connected with exactly four edges.

I created the following quarter sphere by smooth subdiving a cube (edit mode: w > u), shaped it to a sphere (edit mode: Shift + Alt + s) and deleted the excess vertices.





EDIT: After requests and additional info in the question, I will provide a step by step workflow.


Create a cube Shift + A > C, then go into edit mode (Tab). Make sure the pivot stays in the center of the cube.

Do a smooth subdivision operation three times. (W + U)

Select and delete the unnecessary vertices.

Select the outer edge loop (Alt + rightbuttonclick) and scale it to zero along the Y - axis. (S > Y > 0). Make sure to enable proportional editing for this step, so the topology stays even.

With the edge loop still selected, do the to sphere operation.(Shift + Alt + S > 1 > Enter)

Select all vertices A, enable 2D Cursor as Pivot . and do the to sphere operation.(Shift + Alt + S > 1 > Enter), the last edge loop will still be on two axis.

Delete the additional vertices, then select the lower edge loop.

Extrude, scale according to the reference, insert edge loop to make the edges clearer.